In mechanics, friction plays a major role both in the laboratory and industrial worlds. Friction is the resistance to the sliding, rolling, or flowing motion of an object due to its contact with another object. Rolling friction is caused primarily by the interference of small indentations formed as one surface rolls over another. This is the idea behind the frictional forces involved with wheels, cylinders, and spheres. In the ideal case of the wheel, we must first look at the forces acting on the wheel. In pure rolling motion friction is required to start, stop, change the motion of a wheel. Below we can see the frictional force necessary to begin motion, and get the wheel moving at a velocity v. In pure rolling motion, friction causes the wheel to catch and stops the sliding and slipping motion; for example when a car spins its tires, slipping is taking place, thus the frictional force works to stop the spinning out and causes the tires to catch and begin pure rolling motion.
The frictional force, f, the force required to slow the wheel produces a torque which tends to decrease the angular velocity, w. (The normal and gravitational forces produce no effect because their line of action is through the center of rotation.) However, the surface could not possibly have such an effect on the wheel once the wheel has achieved pure rolling motion and constant angular and linear velocity. Zero friction occurs only for horizontal motion at constant velocity, but it is non-zero for any case in which acceleration is occurring parallel to the direction of motion of the center of mass, as when the object is rolling-without-slipping up or down a sloped surface. If we consider the rotation as being about the center of mass of the object, then the frictional force must be in a direction to provide the torque necessary to decrease or increase the angular velocity, depending on whether the object is accelerating or decelerating, respectively. Note that the friction can be in the direction of motion (rolling downhill) or opposite to it (rolling uphill). In pure rolling motion there is no sliding or slipping, thus the contact points have no relative motion (no relative velocity). This results in a frictional force of zero. Therefore, the wheel will roll forward with constant velocity, v = Rw, where R is the radius of the wheel.
In the actual case of the rolling wheel, the free-body diagram is much different. Both the wheel and the surface will undergo deformations due to their particular elastic characteristics. At the contact points, the wheel flattens out while a small trench is formed in the surface. The normal force is now distributed over the actual contact area rather than the point just below the center of the wheel.
Furthermore, the wheel takes on a sort of plowing motion resulting in increased deformation at the front of the wheel, while the rear of the wheel undergoes little deformation which results in the majority of the normal force being located at the front. When the wheel and the surface deform there is a minute amount of slipping, but the majority of the force is due to static friction. The overall rolling friction results in a force at the center of the wheel and is parallel to the surface of contact, and is represented by the equation: